Weak-vertex-pancyclicity of (n, k)-star graphs

The (n, k)-star graph (S-n,S-k for short) is an attractive alternative to the hypercube and also a generalized version of the n-star. It is isomorphic to the n-star (n-complete) graph if k = n - 1 (k = 1). Jwo et al. have already demonstrated in 1991 that an n-star contains a cycle of every even length from 6 to n!. This work shows that every vertex in an S-n,S-k lies on a cycle of length l for every 3 <= l <= n!/(n - k)! when 1 <= k <= n - 4 and n >= 6. Additionally, for n - 3 <= k <= n - 2, each vertex in an S-n,S-k is contained in a cycle of length ranged from 6 to n!/(n - k)!. Moreover, each constructed cycle of an available length in an S-n,S-k can contain a desired I-edge. (C) 2008 Elsevier B.V. All rights reserved.